The population dynamics of a species is governed by the discrete model

$$N_{t+1}=f\left(N_{t}\right)=N_{t} \exp \left[r\left(1-\frac{N_{t}}{K}\right)\right]$$

where $r$ and $K$ are positive constants.

Determine the steady states and their eigenvalues. Show that a period-doubling bifurcation occurs at $r=2$.

Show graphically that the maximum possible population after $t=0$ is

$$N_{\max }=f(K / r) .$$